If Sotheby’s plans to auction a famous Picasso. There are bidders 5 with values distributed uniformly between \$0 and \$50 million.

(a) Suppose that Sotheby’s holds a second-price sealed bid auction with no reserve price. What is the expected revenue? .

(b) The seller proposes to hold an all-pay auction to try to get as much money as possible. In an all-pay auction, each bidder places a bid, the bidder with the highest bid wins the object, but all bidders pay their bids. Bids must be non-negative.

(i) In an all-pay auction what should a bidder with the lowest possible value \$0 bid? .

(ii) In equilibrium, bids in an all-pay auction are a strictly increasing function of a bidder’s value. Explain why this fact, and yours answer to (a) and (b)(i), imply that the expected revenue from the all-pay auction is \$ 33.3 million.

(c) Suppose that Sotheby’s now plans to hold a second-price auction with a reserve price.

(i) What is the optimal reserve price if Sotheby’s knows that there will be 5 bidders? .

(ii) What is the optimal reserve price if Sotheby’s thinks that there will be 5  bidders with probability and bidders with probability

? Explain.

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